Graph Colorings Based On $n$-Dimension Digital-based Strings For Information Security

A new type of total colorings is defined, we call them n-dimension total colorings such that the vertices and edges of graphs are colored with $n$ -dimension digital-based strings $a_{1}a_{2}\cdots a_{n}$ , and hold some restrictive conditions between the colors of vertices and edges of the graphs. We show that All trees admit 2-dimension proper total colorings by the ADDING-leaves algorithm. By trees admitting 2-dimension proper total colorings, we present tree-graphic lattices and 2-dimension Topcode-matrices for building topological cryptography in topological coding. Some mathematical problems for future research are proposed at the end of this article.